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Unformatted text preview: = R ◦ exp ◦ S ◦ T . (A diagram would help to illustrate what f is doing.) It is easy to check that f is conformal, being a composition of conformal maps. More explicitly, we can write f ( z ) = exp ± c za + d ²1 exp ± c za + d ² + 1 . The constants c and d will depend on the initial location of the two circles in the complex plane. ± 2 Conway, Page 67, Problem 6. Show that if γ : [ a,b ] → C is a Lipschitz function then γ is of bounded variation. Proof. Suppose γ has Lipschitz constant M . Let P = { a = t < t 1 < ··· < t m = b } be any partition of [ a,b ]. Then we have v ( γ ; P ) = m X k =1  γ ( t k )γ ( t k1 )  ≤ m X k =1 M ( t kt k1 ) = M ( ba ) . Since P was arbitrary, we conclude that γ is of bounded variation. ± 3...
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This note was uploaded on 10/19/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.
 Three '09
 Cong
 Math

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